# Trigonometry - Geometry - side lengths

posted by Sam

In convex quadrilateral ABCD, angle A is congruent to angle C, AB=CD=180, and AD is not equal to BC. The perimeter of ABCD is 640. Find cos A.

Hmm...I drew the diagram, and labeled some things, I drew an altitude from D to AB and did some other things, but no success was obtained. Any help? Thanks

1. Reiny

joint BD, and use the cosine law twice, once for each triangle
we know AB = DC = 180
let angle A be Ø
let AD = a and BC = b

BD^2 = a^2 + 180^2 - 2(180)a cosØ
BD^2 = b^2 + 180^2 - 2(180)bcosØ

a^2 + 180^2 - 360a cosØ = b^2 + 180^2 - 360b cosØ
a^2 - b^2 = 360a cosØ - 360b cosØ
(a+b)(a-b) = 360cosØ (a-b)
a+b = 360cosØ
cosØ = (a+b)/360

but perimeter = 640
a+b+180+180=640
a+b = 280

then cosØ = 280/360= 7/9

2. Sam

ohhhhh that makes sense i should have thought of that thanks!

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